Chapter 28

What is the de Broglie wavelength associated with a \(1000-\mathrm{kg}\) car moving at \(25 \mathrm{~m} / \mathrm{s} ?\)

If the de Broglie wavelength associated with an electron is \(7.50 \times 10^{-7} \mathrm{~m},\) what is the electron's speed?

An electron and a proton are moving with the same speed. (a) Compared with the proton, will the electron have (1) a shorter, (2) an equal, or (3) a longer de Broglie wavelength? Why? (b) If the speed of the electron and proton is \(100 \mathrm{~m} / \mathrm{s},\) what are their de Broglie wavelengths?

An electron is accelerated from rest through a potential difference of \(100 \mathrm{~V}\). What is the de Broglie wavelength of the electron?

An electron is accelerated from rest through a potential difference so that its de Broglie wavelength is \(0.010 \mathrm{nm} .\) What is the potential difference?

Electrons are accelerated from rest through an electric potential difference. (a) If this potential difference increases to four times the original value, the new de Broglie wavelength will be (1) four times, (2) twice, (3) one-fourth, (4) one-half that of the original. Why? (b) If the original potential is \(250 \mathrm{kV}\) and the new potential is \(600 \mathrm{kV}\), what is the ratio of the new de Broglie wavelength to the original?

A proton and an electron are accelerated from rest through the same potential difference \(V\). What is the ratio of the de Broglie wavelength of an electron to that of a proton (to two significant figures)?

A charged particle is accelerated through a potential difference \(V\). If the voltage were doubled, what would be the ratio of the new de Broglie wavelength to the original value?

A scientist wants to use an electron microscope to observe details on the order of \(0.25 \mathrm{nm}\). Through what potential difference must the electrons be accelerated from rest so that they have a de Broglie wavelength of this magnitude?

According to the Bohr theory of the hydrogen atom, the speed of the electron in the first Bohr orbit is \(2.19 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) (a) What is the wavelength of the matter wave associated with the electron? (b) How does this wavelength compare with the circumference of the first Bohr orbit?

If the absolute value of the wave function of a proton is twice as large at location A than at location B, how many times is it more likely to find the proton in location A than in B?

If you are twice as likely to find an electron at a distance of \(0.0400 \mathrm{nm}\) than \(0.0500 \mathrm{nm}\) from the nucleus, what is the ratio of the absolute value of the wave function at \(0.0400 \mathrm{nm}\) to that at \(0.0500 \mathrm{nm} ?\)

A particle in box is constrained to move in one dimen-sion, like a bead on a wire, as illustrated in \(\mathbf{r}\) Fig. 28.16 . Assume that no forces act on the particle in the interval \(0

(a) How many possible sets of quantum numbers are there for the \(n=1\) and \(n=2\) shells? (b) Write the explicit values of all the quantum numbers \(\left(n, \ell, m_{\ell}, m_{\mathrm{s}}\right)\) for these levels.

How many possible sets of quantum numbers are there for the subshells with (a) \(\ell=2\) and (b) \(\ell=3\) ?

An electron in an atom is in an orbit that has a magnetic quantum number of \(m_{\ell}=2\). What are the minimum values that (a) \(\ell\) and (b) \(n\) could be for that orbit?

Draw schematic diagrams for the electrons in the subshells of (a) sodium (Na) and (b) argon (Ar) atoms in the ground state.

Identify the atoms of each of the following ground state electron configurations: (a) \(1 s^{2} 2 s^{2} ;\) (b) \(1 s^{2} 2 s^{2} 2 p^{3}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6}\) (d) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{4}\)

Write the ground state electron configurations for each of the following atoms: (a) boron (B), (b) calcium \((\mathrm{Ca}),(\mathrm{c}) \mathrm{zinc}(\mathrm{Zn}),\) and \((\mathrm{d}) \mathrm{tin}(\mathrm{Sn})\)

Write the ground state electron configurations for each of the following atoms: (a) boron (B), (b) calcium \((\mathrm{Ca})\) (c) zinc \((\mathrm{Zn}),\) and (d) tin (Sn).

A \(1.0-\mathrm{kg}\) ball has a position uncertainty of \(0.20 \mathrm{~m}\). What is its minimum momentum uncertainty?

An electron and a proton each have a momentum of \(3.28470 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} \pm 0.00025 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) (a) The minimum uncertainty in the position of the electron compared with that of the proton will be (1) larger, (2) the same, (3) smaller. Why? (b) Calculate the minimum uncertainty in the position for each.

If an excited state of an atom has a lifetime of \(2.0 \times 10^{-7} \mathrm{~s},\) what is the minimum error associated with the measurement of the energy of this state?

What is the minimum uncertainty in the speed of an electron that is known to be somewhere between \(0.050 \mathrm{nm}\) and \(0.10 \mathrm{nm}\) from a proton?

What is the minimum uncertainty in the position of a \(0.50-\mathrm{kg}\) ball that is known to have a speed uncertainty of \(3.0 \times 10^{-28} \mathrm{~m} / \mathrm{s} ?\)

The energy of a \(2.00-\mathrm{keV}\) electron is known to within \(\pm 3.00 \% .\) How accurately can its position be measured?

(a) If the lifetime of excited state \(\mathrm{A}\) is longer than that of another excited state \(\mathrm{B}\), then the width of a spectral line due to natural broadening for a transition from state A to the ground state will be (1) smaller than, (2) the same as, (3) greater than that for a transition from state B to the ground state. Why? (b) Calculate the ratio of the width of a spectral line due to natural broadening for a transition from an excited state with a lifetime of \(10^{-12} \mathrm{~s}\) to the ground state to that from a state with a lifetime of \(10^{-8}\) s to the ground state.

What is the energy of the photons produced in proton-antiproton pair annihilation, assuming that both particles are essentially at rest initially?

A muon, or \(\mu\) meson, has the same charge as an electron, but is 207 times as massive. (a) Compared with electron-positron pair production, the pair production of a muon and an antimuon requires a photon of (1) more, (2) the same amount of, (3) less energy. Why? (b) What would be the minimum energy and frequency for such a photon?

An electron traveling at \(3.00 \times 10^{4} \mathrm{~m} / \mathrm{s}\) is further accelerated by a potential difference so as to reduce its de Broglie wavelength to one-third of its original value. How much voltage is required to accomplish this?

A beam of electrons moving at \(5.0 \times 10^{5} \mathrm{~m} / \mathrm{s}\) is incident on a single slit that is \(0.025 \mu \mathrm{m}\) wide. On a screen that is \(1.0 \mathrm{~m}\) behind the slits, an electron diffraction pattern is observed. What is the width of the central maximum?

An electron microscope is an instrument that uses electrons instead of light for the imaging of objects. A monochromatic beam of electrons is accelerated through a potential difference of \(50 \mathrm{~V}\). What is the ratio of the minimum angle of resolution (Section 25.4 ) of this electron microscope compared to a light microscope using light of wavelength of \(550 \mathrm{nm} ?\)

Suppose a starship had a mass of \(1.25 \times 10^{9} \mathrm{~kg}\) and was initially at rest. If its "matter-antimatter engines" produced photons from electron-positron annihilation and focused them to travel backward out from the ship, how many photons would they have to emit to reach \(0.100 \%\) of the speed of light? [Hint: Use conservation of linear momentum and remember that relativity is not needed here. (Why?)]

Using a typical nuclear diameter of \(4.25 \times 10^{-15} \mathrm{~m}\) as its location uncertainty, compute the uncertainty in momentum and kinetic energy associated with an electron if it were part of the nucleus. For energies greater than a few \(\mathrm{MeV}\), particles such as electrons would escape the nucleus. What does this tell you about the likelihood that an electron resides in the nucleus of an atom?

The lifetime of the excited state involved in a He-Ne laser of wavelength \(832.8 \mathrm{nm}\) is about \(10^{-4} \mathrm{~s}\). What is the ratio of the frequency width of a spectral line due to natural broadening to the frequency of the laser?